MoL200113: Lee, Troy (2001) Is Multiplication Harder than Addition? Arithmetical Definability over Finite Structures. [Report]
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Abstract
Is Multiplication Harder than Addition?
Arithmetical Definability over Finite Structures
Troy Lee
Abstract:
In this thesis, we show that two definability results of Julia Robinson,
namely that multiplication and the successor relation can firstorder
define addition, and that the divisibility relation and successor can
firstorder define multiplication, also hold over finite structures (where
ordering is used instead of successor). The first result is obtained by
showing that the BIT predicate can be defined with TIMES and ordering,
thus FO(<,BIT)=FO(<,TIMES). Then the ``carrylookahead'' construction
can be used to define PLUS with BIT. We also show that there is no
firstorder sentence with TIMES but without ordering which can define
PLUS. A corollary is then that TIMES cannot firstorder define ordering.
Our results, together with recent results on the Crane Beach conjecture
show that there is a language definable in FO(<,TIMES) but not in
FO(<,PLUS). In other words, there is no firstorder sentence with < and
PLUS which can express TIMES.
Item Type:  Report 

Report Nr:  MoL200113 
Series Name:  Master of Logic Thesis (MoL) Series 
Year:  2001 
Date Deposited:  12 Oct 2016 14:38 
Last Modified:  12 Oct 2016 14:38 
URI:  https://eprints.illc.uva.nl/id/eprint/726 
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